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Chapter 8: Problem 95

The Pfund series of the hydrogen spectrum has as its longest wavelength component a line at \(7400 \mathrm{nm}\) Describe the electron transitions that produce this series. That is, give a Bohr quantum number that is common to this series.

Short Answer

Expert verified
For the Pfund series of the hydrogen spectrum, the Bohr quantum number corresponding to the transition that produces the longest wavelength of 7400 nm is 6.

Step by step solution

01

Identify given values and unknowns

It is given that the Pfund series' longest wavelength is \( 7400 nm \). This series corresponds to the transitions where the electron ends in the \( n=5 \) energy level of the hydrogen atom. What's common with the Pfund series is the lower energy level, \( n_1 \), which is equal to 5. The higher energy level, \( n_2 \), will be different for each transition and is the one to solve.
02

Solve for the higher quantum number

Substitute the given values into the formula: \[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] Convert \( \lambda \) to meters and solve for \( n_2 \). Remember that \( R \) has a value of \( 1.097373 x 10^7 m^{-1} \).
03

Calculation

After rearranging the formula and substituting the known values, calculate for \( n_2 \). The result should be rounded off to the nearest whole number as quantum number cannot be a fraction or decimal.
04

Interpret the result

The quantum number attained indicates to which energy level the electron transitioned to emit the longest wavelength of light. This is the Bohr quantum number for the given Pfund series transition.

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Most popular questions from this chapter

Identify the orbital that has (a) two radial nodes and one angular node; (b) five radial nodes and zero angular nodes; (c) one radial node and four angular nodes.

What is the length of a string that has a standing wave with four nodes (including those at the ends) and \(\lambda=17 \mathrm{cm} ?\)

When atoms in excited states collide with unexcited atoms they can transfer their excitation energy to those atoms. The most efficient energy transfer occurs when the excitation energy matches the energy of an excited state in the unexcited atom. Assuming that we have a collection of excited hydrogen atoms in the \(2 s^{1}\) excited state, are there any transitions of \(\mathrm{He}^{+}\) that could be most efficiently excited by the hydrogen atoms?

Without doing detailed calculations, indicate which of the following electron transitions requires the greatest amount of energy to be absorbed by a hydrogen atom: from (a) \(n=1\) to \(n=2 ;\) (b) \(n=2\) to \(n=4 ;\) (c) \(n=3\) to \(n=9 ;\) (d) \(n=10\) to \(n=1\)

Without doing detailed calculations, indicate which of the following electron transitions in the hydrogen atom results in the emission of light of the longest wavelength. (a) \(n=4\) to \(n=3 ;\) (b) \(n=1\) to \(n=2\) (c) \(n=1\) to \(n=6 ;\) (d) \(n=3\) to \(n=2\).
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